What is Boolean Algebra?
An algebra in which all elements can take only one of two values (typically 0 and 1, or "true" and "false") and are subject to operations based on AND, OR and NOT
An algebra with two binary operators which are both associative, both commutative, such that both operators are distributive with respect to each other, with a pair of identity elements: one for each operator, and a unary complementation operator which simultaneously yields the inverse with respect to both operators.
Boolean Logic Symbols
Precisely, Boolean Logic is defined as a logical system of operators - 'AND', 'OR' and 'NOT', and is a way of comparing individual bits. These connectors or operators are now used in computer construction, switching circuits, etc.
The AND, OR and NOT operators are also known as logic gates, and are used in logical operation. Their schematic diagram can be viewed from any book based on Boolean Algebra. The following describes the symbols and the operation of the logic gates.
The AND Gate
The AND gate is denoted by a dot (.). In an AND gate, there will be more than one input and only one output. Here, if all inputs are ON, the output will also be ON. And, if either of the inputs is OFF, then the output will also be OFF. The AND gate's symbol is '&'. Lets see the working in an example.
A . B = C (Here, A and B are the inputs, and C is the output)
As we know that in the binary number system, 1 means ON and 0 means OFF. So, if we take the inputs to be 1, the output will also give us 1.
A . B = C
1 . 1 = 1 (A = 1, B = 1).
If any of the input is taken as 0, then output will also be 0
A . B = C
1 . 0 = 0 (A = 1, B = 0)
The OR Gate
The OR gate is denoted by plus (+). Here, there will be more than one input and just one output. If we take both the inputs as 1, the output will also be 1. However, unlike the AND gate, if either of the inputs is 0, the output will still be one. Its symbol is '/'. Example;
A + B = C (Here, A and B are the inputs, and C is the output)
For A = 1, B = 1
A + B = C
1 +1 = 1
For A = 1, B = 0
A + B = C
1 + 0 = 1
For A = 0, B = 0
A + B = C
0 + 0 = 0
The NOT Gate
The NOT Gate is also known as the inverter gate. As the name suggests, here the output will be opposite to the input. There will be one input and one output. That is, if the input is 1 (ON), then the output will be 0 (OFF). The NOT gate is symbolized by a line over top of the input (Ā). The sign is also known as a 'complement'. For example,
For example, if A is the input, the the output will be Ā
That is,
For A = 1, output is 0
And for A = 0, ouput is 1
The NAND and the NOR gates are known to be the universal gates in boolean logic. Their combinations may be used to form any kind of logic gates. A NAND gate is formed by combining a NOT and AND gate. A NOR gate is a combination of a NOT and OR gate. The other logic gates are XOR (exclusive OR) and XNOR gates.
Simplifying Boolean Expressions
For simplifying boolean expressions, there are certain laws which need to be followed. They are the Boolean Rules for Simplification.
The Idempotent Laws
A . A = A
A + A = A
The Associative Laws
(A . B) C = A (B . C)
A + B) + C = A + (B + C)
The Commutative Laws
A . B = B . A
A + B = B + A
The Distributive Laws
A (B + C) = AB + AC
A + BC = (A + B) (A + C)
The Complement Laws
A . Ā = 0
A + Ā = 1
The Involution Law
A(double complement) = A
The Law of Union
A + 1 = 1
A + 0 = A
The Law of Intersection
A . 1 = A
A . 0 = 0
The Law of Absorption
A (A + B) = A
A + (A . B) = A
The Law of Common Identities
A (Ā + B) = A . B
A + (Ā . B) = A + B
DeMorgan's Law
(A . B)(complement) = A(complement) + B(complement)
(A + B)(complement) = A(complement) . B(complement)
Lets take a simple example of a simplified Boolean Expression. Suppose a logical circuit gives an expression A + ĀB = A + B
Now, the simplification goes like this,
A + ĀB
= (A + Ā) (A+B)
= A + B
History
The term "Boolean algebra" honours George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought William Jevons and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward Vermilye Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.
History of Pythagorean Theorem
What is Pythagorean theorem?
The Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypotenuse, or, in mathematical terms, for the triangle shown at right, a2 + b2 = c2. Integers that satisfy the conditions a2 + b2 = c2 are called "Pythagorean triples."
History
There is debate whether the Pythagorean theorem was discovered once, or many times in many places.
The history of the theorem can be divided into four parts: knowledge of Pythagorean triples, knowledge of the relationship among the sides of a right triangle, knowledge of the relationships among adjacent angles, and proofs of the theorem within some deductive system.
Bartel Leendert van der Waerden conjectured that Pythagorean triples were discovered algebraically by the Babylonians. Written between 2000 and 1786 BC, the Middle Kingdom Egyptian papyrus Berlin 6619 includes a problem whose solution is the Pythagorean triple 6:8:10, but the problem does not mention a triangle. The Mesopotamian tablet Plimpton 322, written between 1790 and 1750 BC during the reign of Hammurabi the Great, contains many entries closely related to Pythagorean triples.
In India, the Baudhayana Sulba Sutra, the dates of which are given variously as between the 8th century BC and the 2nd century BC, contains a list of Pythagorean triples discovered algebraically, a statement of the Pythagorean theorem, and a geometrical proof of the Pythagorean theorem for an isosceles right triangle. The Apastamba Sulba Sutra (circa 600 BC) contains a numerical proof of the general Pythagorean theorem, using an area computation. Van der Waerden believes that "it was certainly based on earlier traditions". According to Albert Bŭrk, this is the original proof of the theorem; he further theorizes that Pythagoras visited Arakonam, India, and copied it. Boyer (1991) thinks the elements found in the Śulba-sũtram may be of Mesopotamian derivation.
With contents known much earlier, but in surviving texts dating from roughly the first century BC, the Chinese text Zhou Bi Suan Jing (周髀算经), (The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven) gives an argument for the Pythagorean theorem for the (3, 4, 5) triangle—in China it is called the "Gougu Theorem" (勾股定理). During the Han Dynasty, from 202 BC to 220 AD, Pythagorean triples appear in The Nine Chapters on the Mathematical Art, together with a mention of right triangles. Some believe the theorem arose first in China,where it is alternatively known as the "Shang Gao Theorem" (商高定理), named after the Duke of Zhou's astrologer, and described in the mathematical collection Zhou Bi Suan Jing.
Pythagoras, whose dates are commonly given as 569–475 BC, used algebraic methods to construct Pythagorean triples, according to Proclus's commentary on Euclid. Proclus, however, wrote between 410 and 485 AD. According to Sir Thomas L. Heath, no specific attribution of the theorem to Pythagoras exists in the surviving Greek literature from the five centuries after Pythagoras lived. However, when authors such as Plutarch and Cicero attributed the theorem to Pythagoras, they did so in a way which suggests that the attribution was widely known and undoubted. "Whether this formula is rightly attributed to Pythagoras personally, [...] one can safely assume that it belongs to the very oldest period of Pythagorean mathematics."
Around 400 BC, according to Proclus, Plato gave a method for finding Pythagorean triples that combined algebra and geometry. Circa 300 BC, in Euclid's Elements, the oldest extant axiomatic proof of the theorem is presented.
History of number bases
by Judy Ooi
We have so far come across several different numbering systems, which we can categorize as follows:
- The "Finger" Bases: five, ten, twenty;
- The Binary Bases: two, eight, sixteen;
- Other Bases: twelve and sixty.
Base 2: | ||
A number is even if it ends in 0, odd if it ends in 1 | ||
Base 5: | ||
2: | Any number whose digits add to a multiple of 2 | |
4: | Any number whose digits add to a multiple of 4 |
Base 8: | ||
2: | Any number ending in an even digit | |
4: | Any number ending in 0 or 4 | |
7: | Any number whose digits add to a multiple of 7 |
Base 10: | ||
2: | Any number ending in an even digit | |
3: | Any number whose digits add to a multiple of 3 | |
5: | Any number ending in 0 or 5 | |
6: | Any even number whose digits add to a multiple of 3 | |
9: | Any number whose digits add to a multiple of 9 |
Base 2 (Binary) Numbering system
· Digits used are the Indian/Arabic digits of 0 and 1. Each number occupies a place value. When 1 is reached, the value goes to 0 and 1 is added to the next place value.
o 0,1,10,11,100,101,110,111,1000,etc
· Each place value to the left is equal to 2 times the place value to the right which implies that each place value to the right is equal to the place value to the left divided by 2.
o continuing infinitely <- 256,128,64,32,16,8,4,2,1
Gottfried Wilhelm Leibniz invented the base-two numeration system. He is considered to be amongst the top few intellects in history. He was possibly the last 'universal man' in the sense that he mastered all branches of knowledge known at his time and dealt with them as a unified wholeLeibniz' work in mathematics includes the base-two numeration system and a system of mathematical logic no longer in use. The nature of his thinking is perhaps shown in his comment about imaginary numbers: "The imaginary number is a fine and wonderful recourse of the divine spirit, almost an amphibian between being and non-being." Of course, the invention of calculus will remain as the most lasting contribution of Leibniz to human progress. Leibniz worked out his version of calculus in the three or four years preceding its publication in 1677. The differential symbols -- dx, and dy -- as well as the integral sign ò are due to Leibniz.
Base 5 (Hand) Numbering system
Used primarily before the writing of numbers. The signs or words used are hand for 5, two hands for 10, person for 20 (two hands and two feet). Some cultures would count on fingers with 0 being a closed fist and putting a finger up for one, etc, and some cultures would have 0 being an open hand and 1 would be signified by putting a digit down with 5 represented by a closed fist or 5 down.
Base 5 was used not as a formalized place value, but rather as a grouping value that combined with other values to a larger grouping value. For example 2 fives (two hands) are 10, 4 fives (two hands and two feet-or person) are 20. 12 fives are 60. Each of the groups 10, 20 and 60 were developed into more rigorous numbering systems throughout our globe's cultural history.
Base 8 (Octal) Numbering system
Digits used are the Indian/Arabic numbers 0 thru 7. Each number occupies a place value. When 7 is reached, the value goes to 0 and 1 is added to the next place value.
o 0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17,20,etc
· Each place value to the left is equal to 8 times the place value to the right which implies that each place value to the right is equal to the place value to the left divided by 8.
o continuing infinitely <- 4096,512,64,8,1
· Octal is used to represent binary data. Its use applies primarily to the representation of letters and numbers as they are stored in computers.
Base 10 (Decimal) Numbering system
The Indian culture developed the decimal system. The Mohenjo Daro culture of the Indus valley was using a form of decimal numbering some 5000 years ago. Succeeding cultural changes in this area developed the decimal system into a rigorous numbering system, including the use of zero by the Hindu mathematicians some 1500 years ago. The digits we use for the decimal system are the Arabic/Indian digits of 0 thru 9. Each number occupies a place value. When 1 is reached, the value goes to 0 and 1 is added to the next place value.
Analytic Geometry by Jing Ming
In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions of measurement. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions). Cartesian geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations.
History
The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced cartesian geometry. Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes.
His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.
Analytic geometry has traditionally been attributed to René Descartes. Descartes made significant progress with the methods in an essay entitled Geometry, one of the three accompanying essays published in 1637 together with his Discourse on the Method of Rightly Conducting the Reason in the Search for Truth in the Sciences, commonly referred to as Discourse on Method. This work, written in his native French tongue, and its philosophical principles, provided the foundation for Infinitesimal calculus in Europe.
By Sherina!
History of the Grid System
A typographic grid is a two-dimensional structure made up of a series of intersecting vertical and horizontal axes used to structure content. The grid helps a designer to organize text and images in an easy manner.
Where do you think we got our grid system from? Did it fall from the sky? ; or did some brilliant brain design the system?
Geography consisted of eight volumes. The first discussed the problems of representing a spherical earth on a flat sheet of paper and provided information about map projections. The second was a collection of eight thousand places around the world.
It was then, the brilliant Claudius Ptolemy invented latitude and longitude ; with this, he placed a grid system on the map to locate countries on the map easily. Today, we use the grid system discovered or shall I say invented by Ptolemy. His collection of place names and their coordinates revealed the geographic knowledge of the Roman empire in the second century.
The final volume of Geography was Ptolemy's atlas - featuring maps that utilized his grid system and maps that placed north at the top of the map, a cartographic convention that Ptolemy created. Unfortunately, his gazetteer and maps contained a great number of errors due to the simple fact that Ptolemy was forced to rely upon the best estimates of merchant travelers who were incapable of measuring longitude accurately at that time.
The awesome work of Ptolemy was lost for over a thousand years after it was first published. However, in the early fifteenth century, his work was rediscovered and translated into Latin, the language of the educated populace.
History of Matrix by Balqis :)
Did you know ... ?
The history of matrices goes back to ancient times! But the term "matrix" was not applied to the concept until 1850.
The beginning of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. However it was not until near the end of the 17th Century that the ideas reappeared and development really got underway. The origins of mathematical matrices lie with the study of systems of simultaneous linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive. For example a tablet dating from around 300 BC contains a mathematics problem which involves matrices. The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. An important Chinese text, Nine Chapters of the Mathematical Art (Chiu Chang Suan Shu), gives the first known example of matrix methods to solve simultaneous equations. More uses of matrix-like arrangements of numbers appear in chapter eight, "Methods of rectangular arrays," in which a method is given for solving simultaneous equations using a ‘counting board’ that is mathematically identical to the modern matrix method of solution outlined by Carl Friedrich Gauss (1777-1855), also known as Gaussian elimination.
1 2 3
2 3 2
3 1 1
26 34 39
26 34 39
Our late 20th Century methods would have us write the linear equations as the rows of the matrix rather than the columns but of course the method is identical. Most remarkably the author, writing in 200 BC, instructs the reader to multiply the middle column by 3 and subtract the right column as many times as possible, the same is then done subtracting the right column as many times as possible from 3 times the first column. This gives
0 0 3
4 5 2
8 1 1
39 24 39
Next the left most column is multiplied by 5 and then the middle column is subtracted as many times as possible. This gives
Next the left most column is multiplied by 5 and then the middle column is subtracted as many times as possible. This gives
0 0 3
0 5 2
36 1 1
99 24 39
99 24 39
from which the solution can be found for the third type of corn, then for the second, then the first by back substitution.
 |
| James Joseph Sylvester |
The term "matrix" for such arrangements was introduced in 1850 by James Joseph Sylvester. Sylvester defined a matrix to be an oblong arrangement of terms and saw it as something which led to various determinants from square arrays contained within it. After leaving America and returning to England in 1851, Sylvester became a lawyer and met Cayley, a fellow lawyer who shared his interest in mathematics. Cayley quickly saw the significance of the matrix concept and by 1853, Cayley had published a note giving, for the first time, the inverse of a matrix.
Many theorems were first established for small matrices only, for example the Cayley–Hamilton theorem was proved for 2×2 matrices by Cayley in the aforementioned memoir, and by Hamilton for 4×4 matrices.
Also at the end of the 19th century the Gauss–Jordan elimination (generalizing a special case now known as Gauss elimination) was established by Jordan. In the early 20th century, matrices attained a central role in linear algebra partially due to their use in classification of the hyper complex number systems of the previous century. The inception of matrix mechanics by Heisenberg, Born and Jordan led to studying matrices with infinitely many rows and columns. Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions.
Since their first appearance in ancient China, matrices have remained important mathematical tools. The elevation of the matrix from mere tool to important mathematical theory owes a lot to the work of female mathematician Olga Taussky Todd (1906-1995), who began by using matrices to analyze vibrations on airplanes during World War II and became the torchbearer for matrix theory.
"I did not look for matrix theory. It somehow looked for me." --Olga Taussky Todd in American Mathematical Monthly
